(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
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Require Import RIneq.
Require Import Lia.
Local Open Scope R_scope.

Lemma Rlt_R0_R2 : 0 < 2.
Proof.
change 2 with (INR 2); apply lt_INR_0; apply Nat.lt_0_succ.
Qed.

Notation Rplus_lt_pos := Rplus_lt_0_compat (only parsing).

Lemma IZR_eq : forall z1 z2:Z, z1 = z2 -> IZR z1 = IZR z2.
Proof.
intros; rewrite H; reflexivity.
Qed.

Ltac discrR :=
  try
   match goal with
   |  |- (?X1 <> ?X2) =>
       repeat
         rewrite <- plus_IZR ||
           rewrite <- mult_IZR ||
           rewrite <- Ropp_Ropp_IZR || rewrite Z_R_minus;
       apply eq_IZR_contrapositive; try discriminate
   end.

Ltac prove_sup0 :=
  match goal with
  |  |- (0 < 1) => apply Rlt_0_1
  |  |- (0 < ?X1) =>
      repeat
       (apply Rmult_lt_0_compat || apply Rplus_lt_pos;
         try apply Rlt_0_1 || apply Rlt_R0_R2)
  |  |- (?X1 > 0) => change (0 < X1); prove_sup0
  end.

Ltac omega_sup :=
  repeat
    rewrite <- plus_IZR ||
      rewrite <- mult_IZR || rewrite <- Ropp_Ropp_IZR || rewrite Z_R_minus;
  apply IZR_lt; lia.

Ltac prove_sup :=
  match goal with
  |  |- (?X1 > ?X2) => change (X2 < X1); prove_sup
  |  |- (0 < ?X1) => prove_sup0
  |  |- (- ?X1 < 0) => rewrite <- Ropp_0; prove_sup
  |  |- (- ?X1 < - ?X2) => apply Ropp_lt_gt_contravar; prove_sup
  |  |- (- ?X1 < ?X2) => apply Rlt_trans with 0; prove_sup
  |  |- (?X1 < ?X2) => omega_sup
  | _ => idtac
  end.

Ltac Rcompute :=
  repeat
    rewrite <- plus_IZR ||
      rewrite <- mult_IZR || rewrite <- Ropp_Ropp_IZR || rewrite Z_R_minus;
  apply IZR_eq; try reflexivity.
